The Lorentz force

the lorentz force

How does a charge in motion feel the effect of a Magnetic Field in the surroundings?

The Lorentz Force

Vector Formula

Get a Uniform Magnetic Field

Scalar Formula

The Lorentz Force

The Lorentz Force

To find out the direction of the Lorentz force F you need to apply the Right-Hand-rule

The Right-Hand-Rule

Conventional symbols for vector B

B: OUT B: IN

Virtual lab: a charge in motion in a Magnetic field

The Lorentz Force is always perpendicular to both velocity (v) and displacement (s) of the charge

The work done upon q by the Lorentz Force to move it from A to B, is always zero!

Watch

So, how does a charge in motion feel the effect of a Magnetic Field in the surroundings?

When a charged particle moves through a magnetic field, it experiences a Lorentz Force, providing it is not moving parallel to the field. This force acts at right angles to both the velocity of the particle, v, and the magnetic field, B.

The direction of this force depends on the direction of the velocity of the particle and the magnetic field as well as the sign of the charge of the particle.

From left to right: four examples of the Lorentz force, a simulation, a video about a moving charge in a uniform B.

v not parallel to B

v parallel to B

v perpendicular to B

α= 0°, sin0°=0 Uniform Motion of q

α= 90°, sin90°=1 Uniform Circular Motion of q

0°<α<90° Helical Motion of q (UM +UCM)

The motion of a charge under the effect of a uniform Magnetic Field B, as the angle between v and B changes .

Play these videos!

F=qvB is at right angle with v, then it is a centripetal force: q moves in a Uniform Circular Motion

v perpendicular to B

Flux and Circulation of the Magnetic Field

The previous are two quantities that can be defined for all vector fields, and are useful to describe some important properties of them.

Flux of the Magnetic Field

Given a uniform magnetic field B and a plane surface S, the magnetic field flux through S is given by:

Flux of the Magnetic Field

Gauss Law for Magnetic Fields

Gauss’s law for magnetism states that the total flux through a closed surface must be zero, and this implies that no magnetic monopoles exists

The number of magnetic field-lines which enter a closed surface is always equal to the number of field-lines which leave the surface. In other words: Magnetic field-lines form closed loops which never begin or end.

Gauss Law for Electric Fields

The electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface.

Thus, magnetic field-lines behave in a quite different manner to electric field-lines, which begin on positive charges, end on negative charges.Hence, a positive (negative) charge inside a closed surface generate an Electric field whose flux is positive (negative).

Circulation of the Magnetic Field

We calculate circulation of magnetic field B around a closed path by summing the dot product B ⋅ Δ l for every piece Δ l , treated as vector whose direction is the direction of traversal, and sum them all.

Circulation of B: Ampere Law

Ampere Law

Circulation of B: Ampere Law

Ampere Law

Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element (the Circulation of B) is equal to the permeability times the electric current enclosed in the loop.

Circulation of B and E